Abstract
Letνbe a finite countably subadditive outer measure defined on all subsets of a setX, take a collectionℂof subsets ofXcontainingXand∅, we derive an outer measureρusingνon sets inℂ. By applying this general framework on two special cases in whichν=μ″, one whereμ∈Mσ(𝔏)and the other whereμ∈Mσ(𝔏1),𝔏1⫅𝔏2being lattices on a setX, we obtain new characterizations of the outer measureμ″. These yield useful relationships between various set functions includingμi,μj,μ″, andμ′.
Highlights
We establish a general framework for the study of outer measures associated with a lattice measure, in particular, of two previously studied outer measures: μ which is finitely subadditive and μ which is countably subadditive
Let ν be a finite countably subadditive outer measure defined on all subsets of a set X, take a collection C of subsets of X containing X and ∅, we derive an outer measure ρ using ν on sets in C
We examine two special cases: one in which C is derived from sets related to a lattice L of subsets of X; another in which there are two lattices L1 ⊆ L2 on X and C is derived from L2
Summary
We establish a general framework for the study of outer measures associated with a lattice measure, in particular, of two previously studied outer measures: μ which is finitely subadditive and μ which is countably subadditive. MR(L) is the set of all L-regular measures on A(L) : μ ∈ MR(L) ⊆ M(L) if and only if for any A ∈ A(L), μ(A) = sup μ(L) | L ⊆ A, L ∈ L . For a finitely subadditive outer measure ν, ν is regular if for any A ⊆ X, there exists E ∈ Sν such that A ⊆ E and ν(A) = ν(E).
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